Example - Where is the Potential Zero?

Consider a +3 C charge located 3 cm to the left of a given point. Where can we place a -1 C charge so that the electric potential at the point is zero?

First, let's ask where along the line joining the +3 C charge and the point we could place the -1 C charge to make the potential zero. In how many places can you put the -1 C charge to make V = 0 at the point?

How can we find these points exactly? We know:

• The total potential at the point will be the algebraic sum of the individual potentials created by each charge.
• The potential created by a point charge is given by: V = kQ/r, where
  • Q is the charge creating the potential
  • r is the distance from Q to the point

We need to solve:

k (+3 C) / 3 cm + k (-1 C) / r = 0

When we cancel out the factors of k and C, we get:

+ 3 / 3 cm - 1 / r = 0      r = 1 cm

If you place the -1 C charge 1 cm away from the point then the potential will be zero there. There are two places along the line that will work: 1 cm to the left of the point and 1 cm to the right of the point. (You should verify this using the simulation.)

Are there other places that you could put the -1 C charge to make the potential zero at the point, perhaps not along the line?

The answer we obtained (r = 1 cm) says that all you need to do is place the -1 C charge 1 cm away from the point. There was no reason that it had to be 1 cm to the left or the right of the point. It can in fact be 1 cm in any direction. So there are an infinite number of places that you can put the -1 C charge to make the potential zero: these places form a circle of radius 1 cm centered about the point.

The freedom of not worrying about direction is because potential is a scalar, that is, just a number. The +3 C charge creates a potential (just a number) at the point. The -1 C charge must be placed so that its potential at the point is the negative of that same number. On the other hand, a field has both a magnitude and a direction. If we wanted to ask the same problem as before except that you had to place the -1 C charge to make the electric field zero at the point, then there would only be one place to put it: along the line to the left of the point. (See the electric field Physlab: "Example - is the Field Zero?") This is the only place where the vectors had both the same magnitude and opposite directions.

Where is the potential zero now? Click on the milestone icon to answer a conceptual question that will appear in the milestone window at the upper right. Click the Explanation button to see a detailed solution to the milestone question.

Where is E zero but V non-zero? Click on the milestone icon to answer a conceptual question that will appear in the milestone window at the upper right. Click the Explanation button to see a detailed solution to the milestone question.